Existence and uniqueness results for phase-field systems of grain boundary motion with 3D-crystalline orientation

Abstract

We propose an anisotropic diffuse interface model for three-dimensional grain boundary motion in polycrystals. The model is derived as the $L^2$-gradient flow of an $\mathbb{R}\times {\mathbb{S}}^3$ constrained free energy functional. We include the case of neither smooth nor strictly convex anisotropies, thus allowing for crystalline ones. Our mathematical analysis of the corresponding system of PDEs yields existence of solutions in any fixed time interval as the first main result of the paper. In the second part, we obtain two different uniqueness results, for a particular case of anisotropies, based on some regularity of the solutions to the system. In the case that the domain is one-dimensional, we obtain uniqueness as a consequence of Lipschitz regularity. In the general $N$-dimensional case, we prove local in time uniqueness (and existence) of classical solutions for smooth and compatible initial data.